2022/12/19 by Gregory Arone, Arone, Gregory, Franjo Šarčević +1
Mathematics · #57R40 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Point processes and geometric inequalities #Primary: 57R42 #Secondary: 55R80
paper · pdf · doi:10.48550/arxiv.2212.09809
openalex publication_date 2022/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a manifold M and an integer r>1, the space of r-immersions of M in \mathbb Rn is defined to be the space of immersions of M in \mathbb Rn such that the preimage of every point in \mathbb Rn contains fewer than r points. We consider the space of r-immersions when M is a disjoint union of k m-dimensional discs, and prove that it is equivalent to the product of the r-configuration space of k points in \mathbb Rn and the kth power of the space of injective linear maps from \mathbb Rm to \mathbb Rn. This result is needed in order to apply Michael Weiss's manifold calculus to the study of r-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for r-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of r-immersions.